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Ask The Wizard #420
On average, how many individuals must be chosen at random to ensure that three share the same birthday?
Here is my solution (PDF).
This topic is being explored and debated in my forum located at Wizard of Vegas .
George Wendt, known for his role as Norm on Cheers, passed away on May 20, coincidentally the same date that marked the airing of the last episode of Cheers 32 years earlier. What are the chances of that happening?
Let's rephrase this inquiry to focus on the likelihood of him passing away exactly on the anniversary of the show's finale: the answer is quite straightforward – it's 1 in 365.25. If you were to randomly select two dates from a year, replacing the first selection each time, the chances that both dates would be identical is reflected by the reciprocal of the total number of days in a year, which equals 1/365.25.
Is it possible to approximate the total number of prime numbers below a specific value?
Yes! According to the Prime Number Theorem The count of prime numbers less than n can be estimated using the formula n/ln(n).
According to this mathematical principle, the average spacing between prime numbers from 1 to n can be approximated by ln(n). Therefore, the number of primes below n can be approximated as n/ln(n).
An extension of this mathematical concept suggests that the estimation for the nth prime number is about n*ln(n).
For example, the number of primes under one billion can be estimated as 1,000,000,000/ln(1,000,000,000) = 48,254,942. The actual number of primes under one billion is 50,847,534. So, the estimate is 5.1% too low. This degree of error decreases as n increases.
Navigating downstream with the river's current takes five days by canoe, whereas traveling upstream against the current requires seven days.
- How much time would be needed to paddle the canoe down the river if there was no current?
- How long would it take to drift down the river with the normal current if paddling was not involved?
1. 35/6 days
2. 35 days
Recall that distance = rate × time.
Let define some variables:
- d = distance
- c = rate of canoe
- f = rate of the flow of the river
Let’s represent the two pieces of given information using equations:
- d = (c+f)*5
- d = (c-f)*7
Let's equate the two equations:
(c+f)*5 = (c-f)*7
5c + 5f = 7c - 7f
12f = 2c
c = 6f
If we choose f=1 for our calculations, then c would equal 6 times 1, giving us 6. Additionally, we can calculate d as (6 plus 1) multiplied by 5, which results in 35.
To find the answer to the first query, we must determine t when c equals 0: so, 35 equals 6t.
t=35/6 = 5 5/6 days.
To respond to the second question, we must establish t when c is 0 and f equals 1: thus, 35 equals 1t.
t=35/1 = 35 days.
This topic is also discussed and resolved on the Mind Your Decisions YouTube channel in a video titled Many students at US colleges find this problem difficult to solve. .